Modal expansion of the scattered field: Causality, nondivergence, and nonresonant contribution

Abstract

Modal analysis based on the quasi-normal modes (QNM), also called resonant states, has emerged as a promising way for modeling the resonant interaction of light with open optical cavities. However, the fields associated with QNM in open photonic cavities diverge far away from the scatterer and the possibility of expanding the scattered field with resonant contributions only has not been established. Here, we address these two issues while restricting our study to the case of a dispersionless spherical scatterer. First, we derive the rigorous pole expansion of the T -matrix coefficients that link the scattered to the incident fields associated with an optical resonator. This expansion evinces the existence of a non-resonant term. Second, in the time domain, the causality principle allows us to solve the problem of divergence and to derive a modal expansion of the scattered field that does not diverge far from the scatterer.

Modal analysis of anapoles, internal fields, and Fano resonances in dielectric particles

“…QNM analysis is common in wave physics [40,41] and has attracted a strong interest in photonics when studying resonant light interactions with nanocavities [34,35,[42][43][44][45][46][47][48][49]. QNMs are the eigen-vector solutions of the Maxwell equations without sources and are associated with the complex eigenvalue frequencies p (i) n,α that also correspond to the poles of the scattering operators introduced in the previous section.…”

“…Following the derivation of the scattering operator by Grigoriev et al [34], the modal expansions of the T (i) n and Ω (i) n coefficients were recently derived [33,35]:…”

“…It can be shown that the convergence of this truncated series can be improved for a given truncation order M by approximating the higher order tem α > M [35]. At the first order, the term that is removed when truncating the sum can be approximated by [35]. The contribution from higher order poles |α| > M is then included in the term S (i) nr,n in order to improve the accuracy of the truncated expansion.…”

“…The fact that the strongest field enhancement is observed in absence of anapole condition is properly predicted by our modal analysis. Results are obtained by carrying out a modal analysis of the T-matrix scattering operator in the multipolar Mie basis [33][34][35]. We believe that determining the anapole conditions in high refractive index scatterers in a widely used multipolar theory facilitates the understanding of the origin of this phenomenon.…”

Quasinormal-Mode Non-Hermitian Modeling and Design in Nonlinear Nano-Optics

“…Theoretical QNM formalisms have been initially established for simple and compact resonator geometries (e.g. 1D Fabry-Perot cavities, Mie sphere resonators [23][24][25][26]) in a uniform background, for which analytical expressions of the field are available. It is only recently that complex resonators with different shapes, made of dispersive materials with several possible inclusions (like plasmonic oligomers) or possibly placed in complex environments (e.g.…”